General constraints on dark matter decay from the cosmic
microwave background
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Slatyer, Tracy R., and Chih-Liang Wu. “General Constraints on
Dark Matter Decay from the Cosmic Microwave Background.”
Physical Review D 95.2 (2017): n. pag. © 2017 American
Physical Society
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http://dx.doi.org/10.1103/PhysRevD.95.023010
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PHYSICAL REVIEW D 95, 023010 (2017)
General constraints on dark matter decay from the cosmic
microwave background
Tracy R. Slatyer* and Chih-Liang Wu†
Center for Theoretical Physics, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, USA
(Received 10 November 2016; published 30 January 2017)
Precise measurements of the temperature and polarization anisotropies of the cosmic microwave
background can be used to constrain the annihilation and decay of dark matter. In this work, we
demonstrate via principal component analysis that the imprint of dark matter decay on the cosmic
microwave background can be approximately parametrized by a single number for any given dark matter
model. We develop a simple prescription for computing this model-dependent detectability factor, and
demonstrate how this approach can be used to set model-independent bounds on a large class of decaying
dark matter scenarios. We repeat our analysis for decay lifetimes shorter than the age of the Universe,
allowing us to set constraints on metastable species other than the dark matter decaying at early times, and
decays that only liberate a tiny fraction of the dark matter mass energy. We set precise bounds and validate
our principal component analysis using a Markov chain Monte Carlo approach and Planck 2015 data.
DOI: 10.1103/PhysRevD.95.023010
I. INTRODUCTION
Dark matter (DM) must be stable on time scales
comparable to the lifetime of our cosmos—but it may still
decay with a very long lifetime, a subdominant component
of the DM might decay on shorter time scales, or decays
might transform a slightly heavier metastable state into the
DM we observe today. Such decays are well motivated and
natural in many classes of DM models—including, for
example, R-parity violating decays of the neutralino [1] or
gravitino [2,3], moduli DM [4], axinos [5], sterile neutrinos
[6] and hidden Uð1Þ gauge boson [7]—but are unlikely to
be probed in any terrestrial experiment, due to their very
long time scales. Only indirect searches have the potential
to observe DM decay products; furthermore, only studies of
the early Universe may be able to probe scenarios where a
subcomponent of DM decays with a lifetime shorter than
the present age of the Universe.
Energy injection between recombination and reionization
affects the ionization and thermal history of the Universe
during the cosmic dark ages. Measurements of the cosmic
microwave background (CMB) probe the dark ages, and thus
provide an avenue to constrain any new physics that leads to
such early energy injections: in particular, the nongravitational interactions of DM. DM annihilation or decay during
and after the epoch of last scattering (z ∼ 1000) generically
injects high-energy particles into the photon-baryon fluid; as
these particles cool, they heat and ionize neutral hydrogen,
increasing the residual ionization level after recombination
*
†
[email protected]
[email protected]
2470-0010=2017=95(2)=023010(17)
and hence modifying the CMB anisotropy spectrum, changing the gas temperature history, and distorting the black-body
spectrum of the CMB [8–11]. Consequently, accurate measurements of the CMB by recent experiments—including
WMAP, ACT, SPT and Planck [12–15]—can set stringent
constraints on the properties of DM. In particular, the impact
on the CMB anisotropy spectrum is typically dominated by
annihilation or decay at relatively high redshifts, prior to the
formation of the first stars, where perturbations to the DM
density are small and the astrophysics is simple and well
understood. Consequently, these constraints evade many
uncertainties associated with present-day galactic astrophysics and DM structure formation.
The Standard Model (SM) products of DM annihilation
or decay—which might include gauge bosons, charged
leptons, hadrons, or other exotic particles—in turn decay to
produce spectra of neutrinos, photons, electrons, positrons,
protons and antiprotons. Neglecting the contribution of
neutrinos, protons and antiprotons (see [16] for a discussion
of the latter), for precise constraints it is necessary to
understand the cooling of photons, electrons and positrons,
and their eventual contribution to ionization, excitation
and heating of the gas. Early studies [9,10] used two simple
approximations: (1) that some constant fraction f of the
injected energy was promptly absorbed by the gas, with the
rest escaping, and (2) that the fraction of absorbed energy
proceeding into ionization and excitation is ð1 − xe Þ=3,
whereas that proceeding into heating is ð1 þ 2xe Þ=3,
where xe is the background hydrogen-ionization fraction.
Subsequent studies [17,18] have demonstrated that it is
important to account for delayed energy absorption,
redshift-dependent absorption efficiency, and the fact that
023010-1
© 2017 American Physical Society
TRACY R. SLATYER and CHIH-LIANG WU
PHYSICAL REVIEW D 95, 023010 (2017)
the fraction of deposited energy proceeding into different
channels depends on both the redshift and the energy of the
primary electron/positron/photon.
A recent analysis [19] has presented interpolation tables
describing the power into ionization/excitation/heating from
primary electrons, positrons and photons injected at arbitrary
redshifts during the cosmic dark ages, with initial energies
in the keV–TeV range. This allows easy translation of any
model of annihilating or decaying DM into redshiftdependent source functions for excitation, ionization and
heating. Using these results, [20] studied the impact on the
CMB anisotropy spectrum of keV–TeV photons and eþ e−
pairs produced by DM annihilation. Extending earlier
studies [21–23], that work demonstrated that the imprint
on the CMB anisotropy spectrum was essentially identical
for all models of s-wave DM annihilation with keV–TeV
annihilation products, up to an overall model-dependent
scaling factor, which could be estimated using principal
component analysis (PCA). [20] further provided a simple
recipe for determining the CMB anisotropy constraints on
arbitrary models of annihilating DM: compute the spectrum
of electrons, positrons and photons produced by a single
annihilation, determine the weighted efficiency factor using
the results of [20], and then apply the bound computed by
the Planck collaboration on the product of this efficiency
factor and the DM annihilation cross section.1
In this article, we extend the same approach to the case
of decaying DM. Using the public code CLASS [24], we
compute the effects on the cosmic microwave background
of keV–TeV electrons, positrons and photons injected by
DM decay. Scanning over injection energies and species
defines a set of basis models, which we use as the input to a
PCA. The variance is dominated by the first principal
component, which thus largely describes the shape of the
perturbation to the CMB anisotropy spectrum from arbitrary DM decays. The coefficients of the basis models in the
first principal component trace their approximate relative
impact on the CMB, and hence the “effective detectability”
parameter for photons and eþ e− pairs injected at a range of
different energies. Once the effective-detectability parameters for both a reference DM-decay model and any other
DM-decay model are known, a constraint on the reference
model can be approximately translated to all other models.
We provide the general recipe and results required to
compute effective-detectability parameters for arbitrary
models of decaying DM.
1
Earlier work [21] applied the same principal component
approach to a much broader class of energy injections, with
arbitrary redshift dependence, but that work (a) relied on an
earlier simplified prescription for the energy deposition, and
(b) found that for fully general energy injections, several principal
components were needed to adequately describe the impact on
the CMB. Restricting ourselves to classes of models that can be
described by a single principal component allows for simpler
broad constraints.
We apply the public Markov chain Monte Carlo (MCMC)
code Monte Python [25] to the Planck 2015 likelihood
to compute the precise limit on our reference model, which
we choose (largely arbitrarily) to be DM with a mass of
2 × ð101.5 þ me Þ MeV, decaying to eþ e− pairs (so the
electron and positron each have ∼30 MeV of kinetic energy,
which gives rise to the largest signal as we show later; we later
refer to this reference model loosely as producing 30 MeV
eþ e− ). We compute the MCMC limits for several other
simple models as a cross-check on our effective-detectability
approach, and find good agreement. We provide comparisons of our limits to existing bounds in the literature, finding
that our new constraints are stronger than previous bounds
for sub-GeV DM decaying primarily to eþ e− .
While DM must be stable on time scales longer than the
age of the Universe, a small fraction of the original DM
could decay with a much shorter lifetime, or early decays
from a slightly heavier state might liberate a tiny fraction
of the DM mass energy. We apply the same PCA approach
to decays with lifetimes ranging from 1013 to 1018 s; for
longer lifetimes, the decays occur after the cosmic dark
ages, and the impact on the CMB is indistinguishable
from decays with lifetimes longer than the age of the
Universe. For shorter lifetimes, the decays occur prior to
recombination, and the ionization history is not affected—
we leave studies of the impact on the CMB spectrum for
future work (see also [11,26]). We describe the shift of
the effective-detectability parameters as a function of the
decay lifetime.
In Sec. II, we summarize our methodology for including
the products of DM annihilation and decay in the evolution
equations for the gas temperature and ionization level,
using the public Boltzmann code CLASS. In Sec. III, we
briefly review the essentials of PCA, and then proceed to
derive the principal components in the CMB anisotropy
spectrum induced by DM decay. For our reference model
and several other benchmarks, we then compute constraints
via a full likelihood analysis of the Planck 2015 data, and
present results in Sec. IV. Finally in Sec. V we explain how
to apply our results to constrain arbitrary models of DM
decay, and present examples and comparisons to previous
constraints for various SM final states. We present our
conclusions in Sec. VI. Appendix A contains plots of the
higher principal components; Appendix B describes supplemental data files [27], which are also available at [28].
Throughout this work, we use the cosmological parameters
from Planck 2015 data [15]: Ωb h2 ¼ 0.0223, Ωc ¼ 0.1188,
ns ¼ 0.9667, ln 1010 As ¼ 3.064, τ ¼ 0.066, and 100θs ¼
1.04093.
II. ENERGY INJECTION FROM DARK MATTER
If DM annihilates or decays to SM particles, it will inject
energy into the Universe at a rate given by, for annihilation
and decay respectively,
023010-2
GENERAL CONSTRAINTS ON DARK MATTER DECAY FROM …
ann
dE
hσvi 2 2 2 2
¼
c f X ΩDM ρc ð1 þ zÞ6 ;
dtdV injected
Mχ
dE dec
e−t=τ 2
¼
c f X ΩDM ρc ð1 þ zÞ3 :
dtdV injected
τ
ð1Þ
Here ρc is the critical density of the Universe in the present
day, ΩDM ρc is the present-day cosmological density of cold
DM, and f X is the fraction of the DM (by mass density) that
participates in these decay/annihilation processes, evaluated before the decays/annihilations have significantly
reduced its abundance. Mχ is the DM mass, hσvi is the
thermally averaged cross section for self-annihilating DM,
and τ is the DM decay lifetime. Here we neglect structure
formation; previous studies of the impact of DM annihilation on the CMB anisotropy spectrum have demonstrated
that most of the effect arises from high redshifts, z ∼ 600,
where inhomogeneities in the DM density are small
[21,29]. Energy injection from DM annihilations and
decays extending until late time, and the possible impact
on reionization, is studied in [30–32].
Observable impacts of such injections are controlled by
the absorption of this energy by the gas, and the modification to photon backgrounds. The latter effect is generally
small for models that inject energy during the cosmic dark
ages and are not already excluded [17,26], so we focus on
constraints arising from the former. We refer to absorption
channels, meaning ionization of hydrogen or helium,
excitation or heating of the gas, or distortions to the
CMB spectrum.
The amount of energy proceeding into the different
absorption channels depends on the energy of the primary
injected particle, the redshift of injection, and the background level of ionization at that redshift. Furthermore,
injections of energy at some redshift can lead to energy
absorption at considerably later times, since the time scale
for cooling of photons above a few keV in energy can be
comparable to the Hubble time [9]. Thus computing the
energy absorbed in the various channels requires a fully
time-dependent treatment of the cooling of the annihilation/
decay products, taking into account the expansion of the
Universe. This has been done in the literature [19], for
keV–TeV photons and eþ e− pairs, with results provided as
interpolation tables over injection redshift, redshift of
absorption and energy of the injected primary particle(s).
In general, the CMB signature of an arbitrary model of
decaying/annihilating DM is dominated by the effect of
photons and eþ e− pairs (which may be produced directly in
the annihilation/decay, or subsequently by the decay of
unstable SM annihilation/decay products). The stable final
annihilation/decay products generally also include neutrinos, protons and antiprotons, but neutrinos can be assumed
to escape and the impact of neglecting protons and
antiprotons is rather small [16].
PHYSICAL REVIEW D 95, 023010 (2017)
Consequently, for any given history of energy injection
and spectrum of annihilation/decay products (in the keV–
TeV range), these results can be used to compute the energy
absorbed into each channel as a function of redshift, as
discussed in [19]. It is generally convenient to normalize
this quantity to the total energy injected at the same
redshift. However, for decays with lifetimes much shorter
than the age of the Universe, the rate of energy absorption
may be non-negligible even after the energy injection from
decay has ceased, and so in this case we normalize to the
power that would be injected without the exponential e−t=τ
suppression. Specifically, given the history of energy
absorption into each channel, we define ratio functions
pann=dec;c ðzÞ by
dE ann
¼ pann;c ðzÞc2 Ω2DM ρ2c ð1 þ zÞ6 ;
dtdV absorbed;c
dE dec
¼ pdec;c ðzÞc2 ΩDM ρc ð1 þ zÞ3 :
dtdV absorbed;c
ð2Þ
These ratio functions capture both the model-dependent
parameters controlling the overall rate and the modeldependent redshift dependence; they completely determine
the impact on the CMB. We can also factor out the channelindependent constants to define the channel- and modeldependent efficiency functions f c ðzÞ,
(
f c ðzÞ ≡
−1
pann;c ðzÞðf 2X hσvi
Mχ Þ
annihilating DM;
pdec;c ðzÞ fτX
decaying DM:
The f c ðzÞ functions for annihilation are thus independent
of the overall annihilation rate, and the f c ðzÞ functions for
decay are independent of the decay lifetime if τ ≫ t for all
relevant time scales. These definitions are consistent with
the definition of f c ðzÞ employed by [19] for annihilating
DM, and also with the definitions of fðzÞ for annihilating
and decaying DM employed by [33], only now with the
efficiency function broken down by absorption channel.
The f c ðzÞ functions are obtained by integrating over the
whole past history of energy injection, and depend on both
the DM model and whether it is annihilating or decaying
(as well as the decay lifetime, if it is not long compared to
the age of the Universe).
In Fig. 1 we show the f c ðzÞ curves for c ¼ ionization on
hydrogen, for primary photons and eþ e− pairs, as a
function of injection energy and redshift of absorption.
Different panels show the results for annihilating DM,
long-lifetime (1027 seconds) decay and short-lifetime (1013
seconds) decay.2
2
A species decaying with such a short lifetime would need to
be a subdominant fraction of the DM, or alternatively the decay
might only liberate a tiny fraction of its energy.
023010-3
TRACY R. SLATYER and CHIH-LIANG WU
PHYSICAL REVIEW D 95, 023010 (2017)
redshifts, as the opacity contrast between sub-keV photons
and keV-plus photons becomes more pronounced (see
e.g. [17]). The same structure can be seen in f ion ðzÞ for
injection of photons, at a slightly higher energy; photons in
this energy range dominantly lose energy by Compton
scattering on electrons, and the resulting energetic electrons
go on to produce ionizing photons as discussed above.
Following the standard treatment of recombination [34],
we incorporate the power absorbed into the various
channels as source terms in the recombination equations,
modifying the public CLASS code [24]. CLASS has builtin functionality for including DM annihilation, using a
simplified prescription for the ratio of power absorbed into
different channels; we simply replace this prescription with
our more accurate channel-dependent f c ðzÞ curves.
Specifically, the evolution of the hydrogen-ionization
fraction xe (defined as ne =nH , where nH is the density of
hydrogen and ne is the density of free electrons) satisfies
dxe
1
½R ðzÞ − I s ðzÞ − I X ðzÞ;
¼
ð1 þ zÞHðzÞ s
dz
ð3Þ
where Rs I s are the standard recombination and ionization
rates, and I X the ionization rate due to DM. This last term
has contributions from direct ionization from ground state
H atoms, and from the n ¼ 2 state,
I X ðzÞ ¼ I Xi ðzÞ þ I Xα ðzÞ:
FIG. 1. Effective efficiency function for energy absorption into
hydrogen ionization, for (top) annihilating DM, (middle) longlifetime (1027 seconds) decaying DM, and (bottom) short-lifetime
decaying (1013 seconds) DM, for initial injection of eþ e− pairs
(left) or photons (right) as a function of redshift of deposition and
initial (kinetic) energy of one of the injected particles.
Note the general trend that f c ðzÞ falls at lower redshifts;
this is due to the increased transparency of the Universe as
it expands, leading to more power escaping into photon
backgrounds. The increase in f ion ðzÞ for electron/positron
energies around 1–100 MeV is due to the fact that electrons
in this energy range upscatter CMB photons to
(∼10 eV–keV) energies where they can efficiently ionize
hydrogen; in contrast, for injections of lower-energy eþ e−
pairs, the upscattered CMB photons are too low energy to
contribute to ionization or excitation (and for sufficiently
low energies, the signal becomes dominated by the photons
from annihilation of the eþ e− ). For higher electron energies, the upscattered CMB photons are not efficient ionizers
and move through a universe that is increasingly transparent to them at low redshifts; consequently, the energydependent peak in f ion ðzÞ is more pronounced at lower
ð4Þ
These contributions can be estimated from the energy
absorbed into the hydrogen-ionization (“ion H”) and
excitation (“exc”) channels: they correspond to the number
of additional ionizations per hydrogen atom per unit time.
In terms of the energy absorption rate into these two
channels, we can write
dE ann=dec
1
I Xi ðzÞ ¼
;
dVdt absorbed; ion H nH ðzÞEi
dE ann=dec
1
: ð5Þ
I Xα ðzÞ ¼ ð1 − CÞ
dVdt absorbed; exc nH ðzÞEα
Here Ei ¼ 13.6 eV is the average ionization energy per
hydrogen atom, Eα is the difference in binding energy
between the 1s and 2p energy levels of a hydrogen atom,
and nH ðzÞ is the number density of hydrogen nuclei. The
factor C describes the probability for an electron in the
n ¼ 2 state to transition to the ground state before being
ionized, and is explicitly given by
C¼
1 þ KΛ2s1s nH ð1 − xe Þ
;
1 þ KΛ2s1s nH ð1 − xe Þ þ KβB nH ð1 − xe Þ
ð6Þ
where Λ2s1s is the decay rate of the metastable 2s level, and
K ¼ λ3α =ð8πHðzÞÞ accounts for the cosmological redshifting of Lyman-α photons. HðzÞ is the Hubble factor at
023010-4
GENERAL CONSTRAINTS ON DARK MATTER DECAY FROM …
redshift z, λα is the wavelength of the Lyman-α transition
from 2p level to 1s level, and βB gives the effective
photoionization rates for principal quantum numbers ≥ 2.
Helium ionization follows a similar evolution equation,
but we have neglected the effects of energy injection from
DM on ionization of helium, as generally the fraction of the
injected energy absorbed into helium ionization is small
[19], and the background helium ionization level has little
impact on the recombination history [18].
A fraction of the energy released by DM goes into
heating of the baryonic gas, adding an extra K h term in the
standard evolution equation for the matter temperature T b
(e.g. described in CLASS [24]),
ð1 þ zÞ
dT b 8σ T aR T 4CMB
xe
ðT − T CMB Þ
¼
dz
3me cHðzÞ 1 þ f He þ xe b
2
Kh
−
þ 2T b ;
ð7Þ
3kB HðzÞ 1 þ f He þ xe
with σ T being the Thomson cross section, aR the radiation
constant, me the electron mass, c the speed of light, and f He
the fraction of helium by number of nuclei. The nonstandard term is given by
dE ann=dec
1
Kh ¼
:
ð8Þ
dVdt absorbed; heat nH ðzÞ
Note that throughout this analysis, we employ efficiency
functions f c ðzÞ calculated assuming a standard ionization
history, without additional contributions from exotic
sources of energy injection. We might therefore ask
whether an increased background ionization level could
significantly reduce the fraction of additional injected
energy proceeding into ionization, thus reducing the effect
on the CMB and relaxing the constraints. For models at the
limits of the bounds we present, the modification to the
hydrogen-ionization fraction xe can be as large as Δxe ∼
10% just prior to reionization, which is sufficient to
nontrivially modify the fraction of deposited power proceeding into ionization (vs heating or excitation) for lowenergy electrons [18]. However, a model with even shorter
decay time / larger power injection was considered in [31],
and it was shown in that case that the overall change to the
ionization history from accounting for these backreaction
effects was very small. Furthermore, we expect the CMB
constraint from decaying DM with a long lifetime to arise
mostly from higher redshifts, z ∼ 300 [21], as we discuss
in more depth in the next section. At such redshifts, for
models at the current limit of experimental sensitivity, the
modification to xe is generally at the percent level or lower.
In addition to these arguments, as we discuss later, our
results in this work are in excellent agreement with those
of [35], which includes a correction to the deposition
fractions to approximately but self-consistently include
the effects of the modified ionization history. We defer a
PHYSICAL REVIEW D 95, 023010 (2017)
fully self-consistent treatment of the energy deposition
fractions in the presence of a modified ionization history to
future work.
III. PRINCIPAL COMPONENT ANALYSIS
PCA provides a systematic approach to deriving broadly
model-independent constraints on DM properties. The
effects of energy injection from different DM models on
the CMB anisotropy spectrum are highly correlated, and
consequently can be characterized by a small number of
parameters. PCA yields a basis of principal components
with orthogonal effects on the CMB anisotropies (after
marginalization over the standard cosmological parameters), into which energy injection models can be decomposed; the eigenvalues of these principal components
reflect the detectability of their imprint on the CMB. As
we show, for decaying DM with a given lifetime, the first
eigenvalue generally dominates the others by roughly an
order of magnitude, so that CMB constraints can be
estimated with Oð10%Þ accuracy by considering only
the overlap of a given model with the first principal
component.
We follow the general procedure outlined in [21]; we refer
the reader to that paper for details of our approach. However,
for convenience we summarize the key points below.
We are interested in how different energy injections
change the anisotropies of the CMB after marginalizing
over the standard cosmological parameters. We characterize our basis energy injection models by the following:
(i) species (photons or eþ e− pairs);
(ii) a single energy of injection Ei (in terms of the kinetic
energy of one of the injected particles); where
relevant, it is assumed that Ei ¼ Mχ for annihilation
to photons, and Ei ¼ Mχ − me for annihilation to
eþ e− pairs;
(iii) redshift dependence of the energy injection profile
(annihilation, decay with a lifetime much longer
than the age of the Universe, or decay with a short
enough lifetime to modify the energy injection
profile).
Different basis models are normalized so that f 2X hσvi=Mχ
ðf X =τÞ is held fixed at some value pref for annihilating
(decaying) DM (recall that by f X we mean the fraction of the
DM mass density comprised of the decaying/annihilating
species, and in the case of short-lifetime decays this fraction
is computed before a significant fraction of it decays); the
effect of each model on the CMB is thus fully characterized
by its f c ðzÞ functions (which are determined by the three
factors above).
In general, we hold the redshift dependence of the energy
injection profile fixed, and then generate N basis models
corresponding to different species and energies of injection.
(We perform one analysis where instead we hold the energy
of injection and species fixed, and generate basis models
023010-5
TRACY R. SLATYER and CHIH-LIANG WU
PHYSICAL REVIEW D 95, 023010 (2017)
corresponding to different energy injection profiles.) Using
the modified CLASS code as described in the previous
section, we determine the perturbation to the TT, TE and
EE anisotropy power spectra induced by each basis model,
which we denote ðΔCl Þi for i ¼ 1.:N. The maximum
precision mode in CLASS is turned on for this step, so
that the calculated power spectrum is stable at the 0.01%
level, and we can probe the impact of very small energy
injections. We vary pref and repeat this procedure; this
allows us to (a) test the assumption that the ðΔCl Þi
perturbations are linear with respect to the normalization
factor pref and (b) determine the derivatives ∂ðΔCl Þi =∂pref
in the limit of small pref . For each l and channel, the
derivative is extracted from a polynomial fit, which also
allows us to test the extent of nonlinearity. For the standard
set of six cosmological parameters, the maximum permitted
energy deposition generally lies within the linear regime,
although if the energy deposited is too large, the approximation of nonlinearity eventually breaks down. For 2σ
constraints on DM decay lifetimes we put later, the nonlinearity is within 10%.
These derivatives provide us with the transfer matrix
components,
T li ¼
EE
TE
∂ðΔCTT
l Þi ∂ðΔCl Þi ∂ðΔCl Þi
:
;
;
∂pref
∂pref
∂pref
ð9Þ
Here T li labels the components of the nl × N transfer
matrix T mapping generic energy injections (described in
the space of basis states) into perturbations to the CMB.
Note that each T li is a three-element vector, holding the
perturbations to the TT, TE, and EE anisotropy spectra at
that l.
A generic DM model that annihilates or decays
producing particles in the keV–TeV energy range can
be approximated as a weighted sum over these
basis models (strictly it is an integral; the approximation
is one of discretization), in the sense that—if we assume a
linear mapping between energy injections and perturbations to the CMB anisotropy spectrum—its effect on the
CMB is an appropriately weighted sum of the results for
the basis models. Denoting an arbitrary model as M and
the basis models
P as M i , i ¼ 1.:N, we can schematically
write M ¼ P i αi M i ; more precisely, by this we mean
ðΔCl ÞM ¼ i αi ðΔCl ÞMi . The αi coefficients can be
trivially determined given the spectrum of annihilation/
decay products for M and the DM lifetime or cross
section þ mass; specifically,
8
dN þ −
e e ;γγ
>
hσvi Ei d ln Ei d ln Ei
>
2
<
;
f
X Mχ
Mχ
1
αi ≈
dN þ −
e e ;γγ
pref >
>
: fX 2Ei d ln Ei d ln Ei ;
τ
Mχ
annihilating DM
decaying DM:
ð10Þ
dN
þ −
Here deln eEi;γγ describes the spectrum of eþ e− or γγ pairs
at Ei per annihilation/decay3 (i.e. each member of the pair
has kinetic energy Ei ), and d ln Ei describes the spacing
between the sample energies, which should be chosen to
cover the whole spectra of photon and eþ e− pairs (this is
the discretization approximation), such that
X
dN eþ e− ;γγ
2Ei
d ln Ei
d ln Ei
i
Z
Z
dN γγ
dN eþ e−
≈ 2E
d ln E þ 2E
d ln E;
d ln E
d ln E
ð11Þ
which gives the total energy in photons, electrons and
positrons per annihilation/decay.
For DM that both annihilates and decays with a long
lifetime, the two contributions to energy injection can
simply be added; in our formalism they generally
contribute to different basis models, characterized by
different redshift dependences for the energy injection
history. (For DM that annihilates and decays with a short
lifetime, the redshift dependence of the annihilation is
different to that assumed here, and requires a separate
analysis.)
The perturbation to the CMB anisotropy
due to a general
P
model is then given by ðΔCl ÞM ≈ i αi pref T li ¼ pref T · α~,
where α~ holds the model coefficients describing its overall
normalization and spectrum.
Using the transfer matrix, we can construct the N × N
Fisher matrix Fe as
ðFe Þij ¼
−1
X
X
T Tli ·
·T lj ;
l
ð12Þ
cov
P
where cov is the appropriate covariance matrix for the
anisotropy spectra
X
cov
¼
2
2l þ 1
0
2
ðCTT
l Þ
B
2
×@ ðCTE
l Þ
TE
CTT
l Cl
ð13Þ
2
ðCTE
l Þ
TE
CTT
l Cl
2
ðCEE
l Þ
TE
CEE
l Cl
TE
CEE
l Cl
2
TT EE
ðCTE
l Þ þ Cl Cl
1
C
A:
ð14Þ
For experiments that are not cosmic variance limited
(CVL), we need to include the effective noise power
spectrum. In this work, we use the same noise spectrum
as in [21]. We consider WMAP7, Planck and an experiment
Note that one could also write d ln γγEi ¼ 12 d ln Eγ i (the photon
dN
dN
spectrum) and d lneþEe−i ¼ d lneEþi (the positron spectrum), assuming
charge symmetry.
023010-6
3
dN
dN
GENERAL CONSTRAINTS ON DARK MATTER DECAY FROM …
that is CVL up to l ¼ 2500 for all the anisotropy spectra
we consider (previous studies have indicated that the effect
on the CMB is largest at low to intermediate l values [10]).
The
P effect of partial sky coverage is included by dividing
l by f sky ¼ 0.65. The diagonal elements ðFe Þii describe
the (squared) signal significance per pref for basis model
Mi , before marginalization over the existing cosmological
parameters.
It is critical to marginalize over the standard cosmological parameters, as they can have non-negligible
degeneracies with energy injections [21]. We use
CLASS to study the impact of small variations of the
cosmological parameters and to construct the transfer
matrix from variations in those parameters to changes in
the CMB anisotropies. The full marginalized Fisher
matrix can be constructed as
F0 ¼
Fe
Fv
FTv
Fe
ð15Þ
where Fe is the premarginalization Fisher matrix, Fc is
the Fisher matrix for the cosmological parameters, and
Fv describes the cross terms. The usual prescription for
marginalization is to invert the Fisher matrix, and remove
the rows and columns corresponding to the cosmological
parameters; but when the number of energy deposition
parameters is much greater than the number of cosmological parameters, it is helpful to take advantage of the
block-matrix inversion and write the marginalized Fisher
T
matrix as F ¼ Fe − Fv F−1
c Fv .
Diagonalizing the marginalized Fisher matrix F,
F ¼ W T ΛW; Λ ¼ diagðλ1 ; λ2 ; …:; λN Þ;
ð16Þ
we obtain a basis of N eigenvectors / principal components
e~i , i ¼ 1.:N, by reading off the rows of W, with corresponding eigenvalues λi , i ¼ 1.:N. These principal components lie in the space of coefficients of the basis models,
i.e. they correspond to a set of coefficients of basis models;
while the normalization of the principal components is
rather arbitrary, we choose to normalize them so that in this
space they are orthonormal vectors. We rank the principal
components by eigenvalue, such that e~1 has the largest
eigenvalue.
In general, we can then determine the impact of an
arbitrary model on the Cl ’s, orthogonal to the standard
cosmological parameters, by simply taking the dot product
of its coefficients fαi g with the first PC. Where the first
eigenvalue dominates the variance (i.e. is large compared to
the sum of all other eigenvalues), it can be thought of as a
weighting function, describing the effect of energy injection on the CMB as a function of different injection species
and energies.
To be explicit, let us write
PN an arbitrary model of
decay/annihilation as M ¼ i¼1 αi Mi ðzÞ as above, and
PHYSICAL REVIEW D 95, 023010 (2017)
α~ ¼ fα1 ; α2 ; …αN g. In the Fisher-matrix approximation
(which assumes linearity and a Gaussian likelihood), we
can estimate the Δχ 2 for model M relative to the null
P max
hypothesis of no energy deposition as Δχ 2 ¼ Ni¼1
ð~α ·
e~i Þ2 p2ref λi [since the eigenvalues of the Fisher matrix
describe ðsignificance per pref Þ2 , and hence have units
1=p2ref ], where N max is the number of principal components
we choose to include. From this, we can forecast constraints on decay lifetime; for example, the 2σ limit
corresponds approximately to the constraint,
2
pref < qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
:
PN max
α · e~i Þ2
i¼1 λi ð~
ð17Þ
In particular, for the basis models Mk , where αj ¼ δjk , we
can estimate the constraint on the normalization parameter
(which recall is defined to be f 2X hσvi=Mχ for annihilation,
or f X =τ for decay) to be
2
2
1
pref < qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
:
≲ pffiffiffiffiffi ×
PN max
ð~
e
λ
2
1 Þk
1
λ
ð~
e
Þ
i¼1 i i k
ð18Þ
We see that when the first PC dominates, its component in
the direction of a given basis model is inversely proportional to the constraint on pref for that model, and thus
directly proportional to the constraint on the decay lifetime
τ, for fixed decaying fraction f X . For annihilating DM, the
PC components are inversely proportional to the constraints
on hσvi=Mχ .
In Fig. 2 we show the first PC (after marginalization) for
annihilating, long-lifetime, and (one example of) shortlifetime decaying DM, with a lifetime of τ ¼ 1013 s. Here
we have labeled the various basis models Mi by their
energy of injection and species.
The largest eigenvalue, corresponding to the first PC,
accounts for 97.0% of the variance for long-lifetime decay,
more than 99.9% of the variance for annihilation, and
95.7% of the variance for an example of short-lifetime
decay (τ ¼ 1013 s). Thus in these three cases the first PC
generically dominates the constraints, and we expect an
analysis using only the first PC to give results accurate at
the level of Oð10%Þ. The approximation of dropping later
PCs is much better for the annihilation case, where it is
unlikely to induce even percent-level error.
In this case, therefore, the curves in Fig. 2 directly
map to the strength of the constraint that can be set on
pref by the CMB, or equivalently, the degree to which
injection of particles with a given energy/species dominates any signal in the CMB. The PC for annihilation
closely matches the equivalent f eff curve presented in
[20] (up to an irrelevant normalization factor). We see
that while for annihilating DM no single energy dominates the signal, in the decay case the first PC is peaked
023010-7
TRACY R. SLATYER and CHIH-LIANG WU
e++e-
PHYSICAL REVIEW D 95, 023010 (2017)
e++e-
photons
photons
1.00
1.00
PC1
f(z=300)
WMAP
PLANCK
CVL
0.10
0.10
0.01
4 5 6 7 8 9 10 11 12 4 5 6 7 8 9 10 11 12
Log10[Energy (eV)]
0.01
4 5 6 7 8 9 10 11 12 4 5 6 7 8 9 10 11 12
Log10[Energy (eV)]
e++e-
photons
1.00
FIG. 3. Comparison between the first principal component for
Planck (red line), as described in Fig. 2, and fc ðz ¼ 300Þ for
hydrogen ionization (blue line), as described in Fig. 1.
WMAP
PLANCK
CVL
To confirm our physical understanding of this peak,
note that from the general analysis in [21], we expect
the impact of DM decay on the CMB to be dominated
by redshifts around z ∼ 300. In Fig. 3 we compare the
behavior of the first PC to the f c ðzÞ curve for
hydrogen ionization from DM decay at z ¼ 300, for
the photon/electron/positron energies of our basis
models (i.e. a horizontal slice through the middle
row of Fig. 1). We see that the agreement is excellent.
Similarly, the f eff curve for annihilation [20] is closely
approximated by f c ðzÞ for hydrogen ionization evaluated at z ¼ 600.
We compute the expected constraint on the decay
lifetime, assuming long-lifetime decay, using the first 1–2
PCs; the results are shown in Table I for a range of
injection energies and species (energies refer to kinetic
energies). We can see that including the second PC
0.10
0.01
4 5 6 7 8 9 10 11 12 4 5 6 7 8 9 10 11 12
Log10[Energy (eV)]
e++e-
photons
1.00
WMAP
PLANCK
CVL
0.10
0.01
4 5 6 7 8 9 10 11 12 4 5 6 7 8 9 10 11 12
Log10[Energy (eV)]
FIG. 2. The first principal components for WMAP7, Planck and
a CVL experiment, after marginalization over the cosmological
parameters, for (top) annihilating DM, (middle) long-lifetime
(1027 seconds) decaying DM, and (bottom) short-lifetime (1013
seconds) decaying DM. The x-axis describes the injection energy
(kinetic energy for a single particle) for eþ e− pairs and photons.
around injection of 30 MeV eþ e− pairs for long lifetimes. As mentioned earlier, we attribute this peak to the
high efficiency of ionization by the secondary products
of ∼10–100 MeV eþ e− pairs, and its increased dominance in the case of decaying DM to the fact that the
Universe is more transparent at the lower redshifts
where the signal from decaying DM is peaked (compared to the higher redshifts that provide most of the
signal for annihilating DM models).
TABLE I. Forecast Planck lower bounds on decay lifetime in
units of 1025 s, at 95% confidence, using PCA, for decays to
eþ e− pairs and photons at a range of energies. The species label
Electron refers to decay to an eþ e− pair, with the kinetic energy
of the electron specified in the Energies column. The first column
shows the forecast using only the first principal component, the
second the forecast including the first two principal components.
Species
Energies
PC1
PC1 þ PC2
Electron
10 keV
1 MeV
100 MeV
10 GeV
1 TeV
0.36
0.19
2.49
0.42
0.11
0.37
0.19
2.54
0.45
0.13
Photon
10 keV
1 MeV
100 MeV
10 GeV
1 TeV
0.81
0.15
0.37
0.10
0.11
0.84
0.16
0.41
0.13
0.14
023010-8
GENERAL CONSTRAINTS ON DARK MATTER DECAY FROM …
PHYSICAL REVIEW D 95, 023010 (2017)
∼1013 s are likely to be more tightly constrained by
probes of the Universe’s earlier history, e.g. big bang
nucleosynthesis.
WMAP
PLANCK
CVL
0.10
IV. CONSTRAINTS FROM PLANCK 2015 DATA
0.01
18
17
16
15
Log10[Lifetime (s)]
14
13
FIG. 4. The first principal components for WMAP7, Planck and
a CVL experiment, after marginalization over the cosmological
parameters, for a set of basis models corresponding to energy
injection of eþ e− pairs with injection energy 30 MeV, with
varying decay lifetimes.
changes the constraints by less than 10% in most cases,
although it can be a larger effect [Oð30%Þ] where the
overlap with the first PC is small. This principally occurs
for heavier DM; as we discuss in the next section, these
constraints are most interesting for MeV–GeV DM.
Contributions from higher PCs are negligible.
As mentioned above, we can also choose our basis
models to represent decaying DM with different lifetimes,
but with fixed injection energy and species (and as
previously, fixed pref ). Since the strongest CMB signal
comes from around 30 MeV (in kinetic energy per
particle) electron-positron pairs in the case of decaying
DM with a long lifetime, we fix the injection energy to
this value, consider only eþ e− pairs, and now vary the
decay lifetime between basis models. Repeating the PCA
described above, we find that in this case the eigenvalue
of the first principal component is 98.0% of the total
variance, again dominating the later principal components. This first principal component is shown in Fig. 4,
where now we have labeled the basis models by their
decay lifetimes.
As previously, Fig. 4 can be understood as displaying
the estimated relative strength of constraints from the
CMB on decaying DM with different lifetimes, assuming
the annihilation products are 30 MeV photons or electron-positron pairs. We see that sufficiently short-lifetime
DM is almost irrelevant to the constraints; this is
expected, since decays occurring before recombination
have very little impact on the ionization history. Precise
CMB constraints on such short-lifetime decays are
difficult to obtain, as if we raise f X to the point where
signals from late redshifts can be measured (above
numerical error) in CLASS, there is a very large energy
injection in the early Universe, and linearity certainly
breaks down. Thus this case would require a full likelihood analysis; however, decays with lifetimes less than
The forecast constraints we have calculated so far are
limited by the assumptions of a Gaussian likelihood and
linearity, which are inherent to the Fisher-matrix
approach. To go beyond these assumptions and find
directly the posterior distributions of the cosmological
parameters, including the DM decay lifetime, we use the
publicly available MCMC parameter estimation code
Monte Python, interfaced with CLASS. For the inference
procedure, we use the Planck 2015 temperature (TT),
polarization (EE) and the cross correlation of temperature
and polarization (TE) angular data including three likelihoods: (i) the low-l temperature and low frequency
instrument (LFI) polarization (bflike, 2 ≤ l ≤ 29), (ii) the
high-l plike TTTEEE (30 ≤ l ≤ 2058) likelihood, and
(iii) the lensing power spectrum reconstruction likelihood.
We perform the analysis assuming flat priors on the
following six cosmological parameters ωb , ωc , ns ,
ln 1010 As , τ, and 100θs , as well as a new parameter,
labeled decay, given by the inverse of the DM decay
lifetime in units of s−1 . Our treatment of the energy
deposition is the same as described in the previous
sections. We adopt the Gelman-Rubin convergence
criterion (variance of chain means divided by the mean
of the chain variances), ensuring that the corresponding
R − 1 fell below 0.01. Our constraints and the onedimensional and two-dimensional likelihood contour
plots are obtained after marginalization over the remaining standard nuisance parameters in the Monte Python
package.
In Table II, we give the 95% C.L. lower limit on the
DM decay lifetime, for different injection energies and
TABLE II. Planck lower bounds on decay lifetime at 95% confidence, using a MCMC analysis of actual data, for decays to
eþ e− pairs and photons at a range of energies. The species label
Electron refers to decay to an eþ e− pair, with the kinetic energy
of the electron specified in the Energies column.
Species
Energies
Decay lifetime / 1025 s (95% C.L.)
Electron
10 keV
1 MeV
100 MeV
10 GeV
1 TeV
0.33
0.18
2.31
0.38
0.11
Photon
10 keV
1 MeV
100 MeV
10 GeV
1 TeV
0.74
0.14
0.35
0.11
0.12
023010-9
TRACY R. SLATYER and CHIH-LIANG WU
PHYSICAL REVIEW D 95, 023010 (2017)
FIG. 6. Constraints on DM decay lifetimes from two methods:
the MCMC constraint on decay to 30 MeVelectrons and positrons,
extrapolated to other energies using the first principal component
(red line) and direct MCMC constraints (black crosses).
other. These results are shown explicitly in Fig. 6.
Typically, the true constraints are slightly weaker than
the PCA-based forecasts; this is expected, as nonGaussianity of the likelihood generally reduces significance/weakens constraints [36], and any nonlinearity
also tends to reduce the signal at larger energy
injections.
We thus have confirmed that the first PC can be
used to estimate correct limits on DM decay process.
Furthermore, by calibrating the constraints to those
from the MCMC and using the first principal component to translate the MCMC results to arbitrary models,
we can cancel out most of the difference between the
PCA and MCMC analyses, as we discuss in the next
section.
V. GENERAL CONSTRAINTS ON DM DECAY
FIG. 5. The marginalized posterior probability distributions for
the cosmological parameters (upper panel), and the corresponding two-dimensional joint probability distributions, including
DM decay lifetime τ in units of s. For this example, we consider
decay of DM with mass 2ðme þ 30Þ MeV ≈ 60 MeV, and
assume the only decay channel is to eþ e− .
species. In Fig. 5, we show the one-dimensional and
two-dimensional posterior probability distributions for
the cosmological parameters, in the case where eþ e−
pairs are injected with the electron and positron each
carrying 30 MeV of kinetic energy. Comparing Tables I
and II, we find they are in good agreement with each
As we have shown in Eq. (10), any decaying DM
model can be decomposed into a linear combination of
the basis models with a set of coefficients fαi g, which
in turn can be read off directly from its decay lifetime
and the spectra of photons/electrons produced by its
annihilation. The detectability of any DM model using
the CMB anisotropy spectrum can be estimated by the
dot product of this coefficient vector α~ with the first PC;
conversely, if no signal is seen, this dot product
approximately controls the strength of the constraint
on pref ¼ f X =τ. Truncating Eq. (17) to the first principal
component, we can write the approximate forecast
95% confidence limit as
023010-10
2
1
2
p
pref ≲ pffiffiffiffiffi
¼ pffiffiffiffiffi f ref ;
~ · e~1
λ1 α~ · e~1
λ1 X N
τ
pffiffiffiffiffi
λ1 ~
N · e~1 ;
⇒ τ ≳ fX
2
ð19Þ
GENERAL CONSTRAINTS ON DARK MATTER DECAY FROM …
~ describes the spectrum of photons or electron/
where N
positron pairs (as appropriate to the basis model indexed
by (i) produced in a single decay,
~ ¼
N
dN eþ e− ;γγ
1
;
E
M χ i d ln Ei
i ¼ 1.:N:
ð20Þ
~ should approximate the
The sum over the elements of N
total fraction of the decaying-DM mass that proceeds
into electrons, positrons and photons. (If only a small
fraction of the DM mass decays into electromagnetically
interacting channels, that is naturally captured in this
formalism.)
Most of the discrepancy between the MCMC results
and those of the PCA lies in the overall normalization,
not in the shape of the first PC. Thus we can improve
the PCA-based forecast by performing a single MCMC
analysis for a reference model, and then using the PCA
to predict the relative strength of constraints on other
models. We choose our reference model to correspond
to injection of 30 MeV kinetic-energy electrons and
~ i ¼ δij where j indexes the basis
positrons, i.e. ðNÞ
model corresponding to injection of 30 MeV electrons
and positrons, with a lifetime much longer than the age
of the Universe. Then if the MCMC constraint on this
model for f X ¼ 1 is τ < τ0 , for a general model we can
estimate
τ ≳ fX
~ · e~1
N
τ :
e~1 ð30 MeVeþ e− Þ 0
ð21Þ
In other words, we can use the first principal component
plotted in Fig. 2 to rescale constraints on the DM decay
lifetime obtained from a MCMC analysis of a single
reference DM model (chosen here to be long-lifetime
DM decaying purely to eþ e− pairs with kinetic energy
per positron of 30 MeV). Note that the normalization of
this principal component cancels out; only its shape is
important. In analogy to the f eff parameter defined for
annihilating DM in [20], our detectability parameter geff
for a given model becomes
geff ¼
~ · e~1
N
:
e~1 ð30 MeVeþ e− Þ
ð22Þ
This parameter is proportional to f eff , but has a different
normalization due to the different reference model (the
reference model for f eff was determined by the likelihood analysis already performed by the Planck collaboration, and corresponded to 100% power deposited
into electrons/positrons/photons, with a redshift-dependent but energy-independent fraction of that power
being promptly absorbed as hydrogen ionization). It
is determined by the integral (or discrete sum) of the
PHYSICAL REVIEW D 95, 023010 (2017)
electron and photon spectra weighted by the first
principal component.
For the Planck data, we obtain the MCMC constraint
on our reference model (decay to 30 MeV electrons
and positrons) τ > τ0 ¼ 2.6 × 1025 s at 95% confidence.
(The corresponding PCA forecast limit is 3.25 × 1025 s,
using only the first PC.) Thus for general models we
write
τ ≳ f X geff × 2.6 × 1025 s
ð95% confidenceÞ:
ð23Þ
To validate this approach, in Fig. 6 we compare two
constraints on the DM lifetime for the models presented in Tables I and II: (1) the directly computed
MCMC bounds (Table II), and (2) the MCMC bound
on our reference model, extrapolated to other energies
using the first PC (this is equivalent to rescaling all the
results in Table II by a constant, determined by the
comparison between the MCMC result and PCA forecast for the reference model). In this case, since we are
assuming f X ¼ 1 for all models and considering models which produce only eþ e− pairs or photons at a
specific energy, the bound on the lifetime is directly
proportional to the first principal component (Fig. 2).
We find good agreement, at the ∼10% level, for all
points tested.
We then apply this approach to DM decay to
SM particles, considering 28 decay modes for DM
masses from 10 GeV to 10 TeV; the resulting spectra
of photons and eþ e− pairs are provided in the
PPPC4DMID package [37]. We assume that 100% of
the DM is decaying, with lifetime much longer than the
age of the Universe.
We also provide constraints on DM below 10 GeV
decaying to photons and eþ e− pairs, the latter either as a
direct decay, or via decay to a pair of unstable mediators
(denoted VV) which each subsequently decay to an
eþ e− pair.
The resulting constraints on the lifetime are shown in
Fig. 7. We note several salient points.
(i) The label q ¼ u; d; s denotes a light quark and h
is the SM Higgs boson. The distinction between
polarization of the leptons (left- or right-handed
fermion) and of the massive vectors (transverse or
longitudinal) matters for the electroweak corrections. The last three channels denote models in
which the DM decays into a pair of intermediate
vector bosons VV, which then each decay into a
pair of leptons.
(ii) Decays to neutrinos are the least constrained, and
are only constrained at all at high masses, as the
only photons and eþ e− pairs in these decays are
produced through electroweak corrections (e.g.
final state radiation of electroweak gauge bosons).
These limits are ∼2–3 orders of magnitude
023010-11
TRACY R. SLATYER and CHIH-LIANG WU
PHYSICAL REVIEW D 95, 023010 (2017)
FIG. 8. Lower bounds on the DM decay lifetime, for decay to
eþ e− , from present-day diffuse photon searches (colored lines)
and from our results by using PCA (black crosses) calibrated to
the MCMC bound for our reference model.
FIG. 7. The estimated lifetime constraints on decaying DM
particles, from PCA for Planck calibrated to the MCMC result for
our reference model (injection of 30 MeV electrons/positrons).
The upper panel covers the DM mass range from 10 GeV to
10 TeV. The lower panel covers the range from keV-scale DM
masses up to 10 GeV for the eþ e− , γγ and VV → 4e channels.
weaker than present-day indirect searches using
neutrino telescopes [38].
(iii) Other SM final states populate a band of decaylifetime constraints whose vertical width is roughly a
factor of 4–5.
(iv) In contrast to annihilating DM, the detectability
function is quite sharply peaked around ∼100 MeV
electrons/positrons, for decaying DM. Consequently,
channels that produce copious soft electrons/
positrons can have enhanced detectability—this
is in contrast to the usual situation for indirect
searches in the present day, where softer spectra
are typically more difficult to detect due to larger
backgrounds.
(v) For TeV DM and above, the contributions from the
electron/positron and photon spectra are typically
comparable, and the detectability depends primarily
on the total power proceeding into electromagnetic
channels.
One might ask how these constraints compare to existing
bounds. For long-lifetime decaying DM, there are stringent
constraints on the decay lifetime from a wide range of
indirect searches (e.g. [39–48]). In general, these constraints
are considerably stronger than our limits, probing lifetimes
as long as 1027−28 s. The exception is for MeV–GeV DM
decaying to eþ e− pairs; these pairs are difficult to detect
directly. They do produce photons via internal bremsstrahlung and final state radiation, and in [47], data from
HEAO-1, INTEGRAL, COMPTEL, EGRET, and the
Fermi Gamma-Ray Space Telescope (Fermi) were used to
set constraints on such decays by searching for these
photons. These constraints are conservative in that they
subtract no astrophysical background model, but they do
assume a Navarro-Frenk-White [49] density profile for
the DM.
In Fig. 8 we compare our CMB constraints (which are of
course independent of any assumptions about the halo DM
density) to these limits. In the MeV–GeV mass range, our
limits exceed the previous best bounds on the decay
lifetime by a factor of a few.
One might ask how much these bounds have the
potential to improve. As shown in Fig. 2, the shape of
the first PC is very similar for WMAP7, Planck and an
experiment that is CVL up to l ¼ 2500. Thus the main
effect of moving closer to a CVL experiment is to
improve the constraints on all channels by a constant
factor. Examining the eigenvalues of the first PC in the
Planck and CVL cases, we expect the limit to improve
by a factor of ∼5 with an experiment that is CVL up
to l ¼ 2500.
Let us now discuss the case where a small mass
fraction of the DM decays prior to the present day. This
immediately removes most limits from present-day
023010-12
GENERAL CONSTRAINTS ON DARK MATTER DECAY FROM …
PHYSICAL REVIEW D 95, 023010 (2017)
1026
PC1
PC1-5
Normalized
τ / fX (s)
1025
1024
1023
18
indirect searches. Limits from structure formation, in the
case where the decay is from a metastable excited state
of DM and thus confers a velocity kick on the remaining DM, can constrain decays with lifetimes ∼1016 s
(e.g. [50–53]). At lifetimes much shorter than
∼1012–13 s, limits from big bang nucleosynthesis generally dominate (for one example scenario, see [54]).
However, in the lifetime range ∼1013–16 s, limits from
the CMB are uniquely powerful [33].
In this case, our PCA must be extended to account for
shorter lifetimes. Figure 4 allows us to approximately
translate the MCMC limits on long-lifetime DM,
decaying to 30 MeV electrons/positrons, into limits
on the same decay channel but for shorter lifetimes.
We can then perform PCA holding the lifetime fixed but
varying the energy of injection and injected species, as
in the case of long-lifetime DM, to translate these
bounds into limits on other channels at the same
lifetime. We show the resulting e~1 curves in Fig. 9.
The analogous curves can be obtained for intermediate
lifetimes by interpolation.
It is interesting to note that as the decay lifetime
becomes shorter, the first PC comes to resemble that
for annihilation; this is because the difference in the
PCs between long-lifetime decay and annihilation arises
from the different redshifts at which the main contribution to the signals occurs. As the decay lifetime is
shortened, more of the signal originates from higher
redshifts, and the PC for decay becomes more similar
to that for annihilation (a redshift of 600, where the
contribution to the annihilation signal peaks, corresponds to a cosmic age of ∼3 × 1013 s).
It is worthwhile to note that if the first PC is
suppressed, the high PCs could give a sizable contribution to the forecast constraint. We show in Fig. 10
the constraints obtained by using different numbers of
16
15
Log10[Lifetime (s)]
14
13
FIG. 10. The forecast bound on decaying DM properties,
obtained by using different numbers of PCs. The red line uses
the first PC and blue line uses the first 5 PCs; the contribution
from higher PCs is negligible. The green dashed line is the
result of the blue line normalized to the MCMC result for a
lifetime of 1018 s. MCMC results for shorter lifetimes are
shown with black crosses, as a cross-check on the PCA-based
extrapolation.
PCs. For the short-lifetime DM, the correction from the
high PCs becomes important. The MCMC results in
this plot show that the difference between the PCA
prediction and MCMC result is not a constant ratio
with respect to lifetime. We therefore normalized the
PCA result (summing up the higher PCs) to the
MCMC result for τ ¼ 1018 s, and used this normalized
curve to estimate the constraints on short-lifetime
decays. The resulting constraint is slightly weaker than
we would obtain using the full MCMC for the shortest
lifetimes we test; thus our constraints on short lifetimes
are conservative.
In Fig. 11 we show the resulting estimated bounds
on the mass fraction of DM that can decay, as a
100
Maximum allowed mass fraction
FIG. 9. First principal components for Planck for annihilation
(red), decays with fixed short lifetimes of 1013 (blue), 1014
(green) or 1015 s (purple), and long-lifetime (1027 s) decays
(gray).
17
10-2
10-4
10-6
10-8
10-10
10-12
1014
1016
1018
1020
Lifetime (s)
1022
1024
1026
FIG. 11. Range of upper bounds on the mass fraction of DM
that can decay with a lifetime τ, for injections of 10 keV–10 TeV
photons and eþ e− pairs; the width of the band represents a scan
over injection species and energy. The constraint is based on the
PCA (first PC only) calibrated to the MCMC bound for our
reference model.
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PHYSICAL REVIEW D 95, 023010 (2017)
function of decay lifetime, based on the 2015 Planck
data. Rather than show results for individual models
(which would require a scan over DM mass and
annihilation channel), we simply show the band traced
out by injection of eþ e− pairs and photons at 10 keV–
10 TeV energies. This figure updates Fig. 8 of [33].
Note that our limits weaken more rapidly than the
bounds in [33] as the decay lifetime becomes shorter
than the age of the Universe at recombination (i.e.
∼1013 s); we attribute this to the fact that [33] used an
older prescription for the fraction of power proceeding
into ionization, which significantly overestimated the
power into ionization when the background ionization
level is non-negligible (as is the case during and
shortly after recombination) [18].
Recently, a complementary analysis using similar
tools was presented in [35]; that work does not use
PCA to set general constraints as we have done, but
explores CMB limits on both decaying DM and other
relics, including primordial black holes, as well as a
range of other non-CMB constraints on short-lifetime
decays. We have compared our results shown in
Fig. 11 to the analogous result from [35]; at first
glance, our constraint band appears to be slightly
broader (with both stronger and weaker limits at most
lifetimes, depending on the DM mass), with the most
marked differences occurring for decay to electrons
(rather than photons). The constraints of [35] are also
somewhat stronger than the ones presented in this work
for low lifetimes τ ≲ few ×1013 s; this is to be
expected, since as discussed above our bounds on
short lifetimes are conservative.
There are several small differences between the
analysis methods in the two works. The authors of
[35] rely on full MCMC runs, rather than only using
a single MCMC to calibrate the PCA, and so can
only test a limited number of DM masses and
lifetimes. Furthermore, they adjust the f c ðzÞ functions
to attempt to self-consistently approximate the effects
of the modified ionization history on the energy
absorption.
However, despite these differences, we find excellent
agreement between the two results when the same set
of Planck likelihoods is used. The default analysis of
[35] does not include the low-l temperature/polarization
likelihood from Planck; instead the authors impose a
prior on τreio taken from Planck 2016 results. Making
the same choice in our analysis narrows the band in
Fig. 11 to closely match that of [35].
VI. CONCLUSION
Using principal component analysis, we have demonstrated that the imprint of general models of decaying
DM on the CMB anisotropy spectrum—via changes to
the ionization and temperature history—can be approximately described by a single parameter, if the lifetime
of the DM is much longer than the age of the Universe.
After performing a detailed likelihood analysis on a
single model to calibrate the constraints, which we have
done using Planck 2015 data, limits on the decay
lifetime for all other models can be determined by a
simple integral of the photon/electron spectra from
annihilation products, weighted by the first principal
component. Including higher principal components
changes the decay lifetime constraints by less than
10% in most cases, and we have validated our approach
with MCMC studies.
We find lifetime constraints typically of the order of
1025 s. These constraints outperform limits from the
galactic diffuse emission for MeV–GeV DM annihilating primarily to eþ e− pairs (or to particles which
decay dominantly to eþ e− ). More generally, they
provide a robust limit on decay lifetime for a very
wide range of models, evading any uncertainties associated with astrophysical backgrounds or the DM
density distribution.
We can also constrain the decay of a subdominant
DM species, or a metastable state of DM, with lifetimes
much shorter than the current age of the Universe, so
long as the lifetime exceeds ∼1013 s (roughly the age of
the Universe at recombination). For shorter lifetimes, the
constraints weaken drastically, and numerical issues
limit our ability to compute even these weakened
bounds; it is likely that for lifetimes much shorter than
1013 s, constraints from distortions of the CMB energy
spectrum or modifications to big bang nucleosynthesis
will be stronger than those computed with our current
approach.
For such short lifetimes, only a tiny fraction of the total
mass density of DM can decay, either because each decay
liberates only a small fraction of the original particle’s
energy, or because the decaying species is only a small
fraction of the total DM. We set upper limits on the mass
fraction of DM that can decay as strong as 10−11 , for
lifetimes ∼1014 s.
ACKNOWLEDGMENTS
We thank Hongwan Liu, Lina Necib, Nicholas Rodd,
Ben Safdi, and Wei Xue for helpful discussions,
Bhaskar Dutta for a conversation that stimulated this
project, and the anonymous referee for valuable comments on the manuscript. This work is supported by the
U.S. Department of Energy under Awards No. DESC00012567 and No. DE-SC0013999. C.-L. W. is partially supported by the Taiwan Top University Strategic
Alliance (TUSA) Fellowship.
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GENERAL CONSTRAINTS ON DARK MATTER DECAY FROM …
e++e-
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PHYSICAL REVIEW D 95, 023010 (2017)
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PC1
PC2
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4 5 6 7 8 9 10 11 12 4 5 6 7 8 9 10 11 12
Log10[Energy (eV)]
4 5 6 7 8 9 10 11 12 4 5 6 7 8 9 10 11 12
Log10[Energy (eV)]
photons
PC1
PC2
PC3
0.4
0.2
4 5 6 7 8 9 10 11 12 4 5 6 7 8 9 10 11 12
Log10[Energy (eV)]
PC1
PC2
PC3
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0.2
photons
photons
PC1
PC2
PC3
4 5 6 7 8 9 10 11 12 4 5 6 7 8 9 10 11 12
Log10[Energy (eV)]
FIG. 12. The first three principal components for WMAP7 (left), Planck (middle) and a CVL experiment (right), for annihilating DM
(top), decaying DM with a long lifetime (middle) and decaying DM with a lifetime of 1013 s (bottom).
APPENDIX A: SUBSEQUENT PRINCIPAL
COMPONENTS
In Fig. 12 we display the second and third principal
components, in addition to the first PC displayed in the
main text.
APPENDIX B: SUPPLEMENTAL MATERIAL
We make available .fits and .dat files [28] containing the
values of the curves plotted in Figs. 9 and 10. We also
provide a Mathematica notebook to demonstrate the use of
these files, with a worked example for how to compute the
Planck constraints on the mass fraction of DM decaying to
muons, for different lifetimes.
There are two .fits files, each one with a corresponding
.dat file.
(i) EnergyPC1: This file contains the results plotted in
Fig. 9, containing arrays of the first PCs for
annihilating DM and decaying DM with lifetimes
1013 ; 1013.5 ; 1014 ; 1014.5 ; 1015 , and 1027 seconds.
The first column (labeled “log10energy”) gives
the base 10 log of the (kinetic) energy in eV of
one of the particles in the pair, the second column
(labeled “ann”) gives the result for annihilation, and
subsequent columns give the results for decay with
lifetimes 1013 (“decay13”), 1013.5 (“decay135”),
1014 (“decay14”), 1014.5 (“decay145”), 1015 (“decay15”), and 1027 (“decay27”) seconds. The first 41
entries correspond to injection of eþ e− pairs, and the
following 41 entries to injection of photons.
(ii) lifetimePCA: The arrays in this file give the
results of the PCA considering injection of
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PHYSICAL REVIEW D 95, 023010 (2017)
þ −
30 MeV e e pairs and varying the decay
lifetime, as plotted in Fig. 10. The first column
(labeled “log10lifetime”) gives the base 10 log
of the decay lifetime in seconds. The second
column (“PC1”) gives the Fisher-matrix forecast
constraint using only the first PC, the third
column (“PCsum”) gives the forecast constraint
using the sum of the first five PCs, and the
fourth column (“Normalized”) gives the forecast
constraint based on the first five PCs, normalized
so that for long lifetimes it matches the MCMC
result.
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